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An integral representation of the Bergman-Weil type is derived for a function defined on an algebraic variety. This formula is useful for constructing Mandelstam-type integral representations for N-particle functions.


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This is a library Circulating Copy which may be borrowed for two weeks. For a personal retention copy, call Tech. Info. Diuision, Ext. 5545 The problem of obtaining a Mandelstam-type representation for an N-particle function has three parts. First one must define the physical sheet· of this function and identify its boundary cuts. Some ,2,3 . . work has been done on this, and more is in progress. Having identified the boundary cuts, one must obtain formulas for the discontinuities and multiple discontinuities across these cuts. . ,4 6 Considerable work' 5, has been done on this, and more is in progress.
Having identified the cuts and obtained formulas for their discontinuities. one must finally express the function on the physical sheet in terms of these discontinuities. This last problem is the one discussed here~ That is, we assume that the cuts bounding the physical sheet and their discontinuities are known, and seek a representation of the function on this sheet.
The Bergman-Weil representation 7 can be used to represent a () function in terms of its multiple discontinuities. Indeed~ the representation given by Mandelstam is essentially a special case of theB-W formula. For the general N-particle function· one needs, UCRL-1786l -2however, a generalization of the B-W formula to the case in which the function is defined only on an algebraic variety. The reason for this is discussed next.
The N-particle function is originally defined only on the mass shell 7Yt, Which is the set of points in momentum-energy space that satisfy the mass constraints and momentum-energy conservation laws.
Since these conditions are expressed by the simultaneous vanishing of severalan?-lytic functions of the momentum-energy vectors, the set·'77/ is an analytic variety. Since these functions are in fact polynomials, "'lis moreover an algebraic variety. The B-W formula refers to functions defined over a full space and hence is not immediately globally. applicable to a function defined only on the algebraic variety· 1Yf .
One can try to avoid this difficulty by invoking Lorentz invariance and reducing the problem to a corresponding problem in the space of scalar invariants~ This works fine for N == 4 and N == 5.
But for N> 5 it doesn't. In particul~r, for N > 5 there is no choice of independent scalar invariants such that all others can be expressed in terms of them as single-valued functions. 'l'h:i.,s means that the mass shell 1rt maps into a multisheeted surface over any space·· J of independent scalars.
One can apply the B':'W formula to the. function defined on an individual sheet. However, such a sheet corresponds to only part of the mass sh~ll ~ • Accordingly it is bounded in part by cuts that .are not the images of cuts that bound the physical sheet in 7l; . Ii . ,

UCRL-1786l
-3-These extra cuts, which arise solely from the multisheeted nature of the image of itt, are "kin~maticaltl, in that they have no counterpart in i 17 itpelf, and depend on the particular choice of independent scalars. Their discontinuities are not given by unita:i-ity, and hence, in distinction to those for the cuts in i?7' must not be considered to be given.
In simple cases one can eliminate the unknown discontinuites across these kinern8. tic, cuts by considering all the sheets simultaneously.
However, the algebra becomes intractable for all but the simplest cases, because one needs solutions of fifth-order algebraic equations.
In order to resolve this problem one can regard the mass shell in invariant space not as a multisheeted surface over some space ~Of independent scalar invariants, but rather as an algebraic variety  It should be emphasized that the variety V is not required to be an analytic manifold. In particular V can have points P such that no neighborhood of P is topologically equivalent to a.  1 1 i=l,·· ·N, The B-W formula'7 gives, formally, for z E V in the interior of iR(a, E, p), fez) The symbol 7 labels the various subsets consisting of n indices from among the set of N+m+n indices ( i ···i· i ···i· i ···i) l' N' l' m' l' n· Thus N' + N" + N'" = n for all 7. The polynomial J (z' ;z) is the deterr r r r minant of the matrix of polynomials Pji(z';z) defined in reference '7.

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The region of integration N' -9- The proof of the theorem consists of four parts. First it is shown that the parameters 0, E, and p can be constrained in such a way that the right side of (2.7) is well defined, and equals the left side.
The limit E ~O is then studied and it is shown that the factors is vlell defined when ~(z,) is well defined. In this case one has, by definition, where the sign is determined by the orientation of N(Z'O)' To establish conditions undE;'r which the requisite functions xi(t). and Yi(t) exist, consider the real polynomial mapping r : R 2n -7 R n ,de fined by be the set of points w E R 2n such that the rank of the n by 2n "" matrix II or jow II is less than n. According to Sard's theorem where q is finite, and the P. are polynomials that are not identically zero: In place of the variable r = (5 i ; E. i; P i) we introduce For any (E, 5) in QnN the rank of II dr/dwll is n at every point . This is what has just been shown. But if the rank of this matrix is n, then the set of n real variables r. can be augmented locally' 1 by a set of n real variables ti to give a set of 2n real variables such that the 2n-by-2n inatrixlldr/dw; dt/dWII has rank 2n.
We wish now to take the limit E ~O in (2.7). To evaluate the result, (2.7) is first converted to an alternative form. .1 For a g~ven term , the denominator in (2.7) can be written as  The equivalence of the contributions to the right sides of (2.21) and (2.7) from the region J (z') ~ E' for any E' > 0, and the absolute , convergence of the right side of (2.7), ensure that the right side of (2.21) exists as a Lebesgue integral.  consider it to be defined in the mean as limit E ~O of integrals. The question of how fast is "sufficiently fast" is, of course; important. Evidently the general formula will have terms with fewer denominators than appear in (2.1), just as in the case of the Mandelstam representation. However, this question is not embarked upon here, where the concern has been with contributions from the singular points of the variety, rather than points at infinity.
The integral (C.3) vanishes due to the constraint Z ij Proof: By a rectifiable curve we mean a curve can be considered to be limit of a a/set of straight-line" segments. Consider the contribution from a set of n such segments. By an adjustment of the definition of the Zi's, these segments can be made to be se@nents of the real axis.